Deltoidal icositetrahedron

[3] The 24 faces are kites. The short and long edges of each kite are in the ratio 1:(2 − 1/√2) ≈ 1:1.292893...

If its smallest edges have length a, its surface area and volume are

${\displaystyle {\begin{aligned}A&=6{\sqrt {29-2{\sqrt {2}}}}\,a^{2}\\V&={\sqrt {122+71{\sqrt {2}}}}\,a^{3}\end{aligned}}}$

Where the short and long edges meet creates an angle of about 81.58°. The exact value is below represented by theta.

${\displaystyle {\begin{aligned}\theta =\arcsin {\sqrt {{\dfrac {5}{8}}\,+{\dfrac {1}{2{\sqrt {2}}}}}}\end{aligned}}}$

The deltoidal icositetrahedron is a crystal habit often formed by the mineral analcime and occasionally garnet. The shape is often called a trapezohedron in mineral contexts, although in solid geometry that name has another meaning.

The great triakis octahedron is a stellation of the deltoidal icositetrahedron.

The deltoidal icositetrahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron.

When projected onto a sphere (see right), it can be seen that the edges make up the edges of an octahedron and cube arranged in their dual positions. It can also be seen that the threefold corners and the fourfold corners can be made to have the same distance to the center. In that case the resulting icositetrahedron will no longer have a rhombicuboctahedron for a dual, since for the rhombicuboctahedron the centers of its squares and its triangles are at different distances from the center.

This polyhedron is topologically related as a part of sequence of deltoidal polyhedra with face figure (V3.4.n.4), and continues as tilings of the hyperbolic plane. These face-transitive figures have (*n32) reflectional symmetry.

• Deltoidal hexecontahedron
• Tetrakis hexahedron, another 24-face Catalan solid which looks a bit like an overinflated cube.
• "The Haunter of the Dark", a story by H.P. Lovecraft, whose plot involves this figure
• Pseudo-deltoidal icositetrahedron

• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208 (The thirteen semiregular convex polyhedra and their duals, Page 23, Deltoidal icositetrahedron)
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 286, tetragonal icosikaitetrahedron)

• Eric W. Weisstein, Deltoidal icositetrahedron (Catalan solid) at MathWorld.
• Deltoidal (Trapezoidal) Icositetrahedron – Interactive Polyhedron model